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Conditionally free reduced products of Hilbert spaces

Volume 254 / 2020

Octavio Arizmendi, Miguel Ballesteros, Francisco Torres-Ayala Studia Mathematica 254 (2020), 23-44 MSC: Primary 46L53; Secondary 46L09. DOI: 10.4064/sm190207-11-7 Published online: 6 March 2020


We present a product of pairs of pointed Hilbert spaces that, in the context of Bożejko, Leinert and Speicher’s theory of conditionally free probability, plays the role of the reduced free product of pointed Hilbert spaces, and thus gives a unified construction for the natural notions of independence defined by Muraki.

We additionally provide important applications of this construction. We prove that, assuming minor restrictions, for any pair of conditionally free algebras there are copies of them that are conditionally free and also free, a property that is frequently assumed (as hypothesis) to prove several results in the literature. Finally, we give a short proof of the linearization property of the $^cR$-transform (the analog of Voiculescu’s $R$-transform in the context of conditionally free probability).


  • Octavio ArizmendiCentro de Investigación en Matemáticas
    Jalisco s/n, Mineral de Valenciana
    Guanajuato, Gto., C.P. 36240, México
  • Miguel BallesterosMathematical Physics Department
    Instituto de Investigaciones
    en Matemáticas Aplicadas y Sistemas
    Universidad Nacional Autónoma de México
    Campus C.U., Circuito Escolar 3000
    C.P. 04510, México City, México
  • Francisco Torres-AyalaDepartamento de Matemáticas
    Facultad de Ciencias
    Universidad Nacional Autónoma de México
    Campus C.U., Circuito Exterior s/n
    C.P. 04510, México City, México

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